depends what you mean?

HRT names do seem like Iterator names. a lot of them follow the Adjective Noun format (Jovial Merryment, Superstitional Realism, Comely Material Morning)

Real life Horse names seem more like... Well, they have a Family name and a Given name, at least the Japanese ones.

Air Groove (part of the Air family), Agnes Tachyon (part of the Agnes family), Narita Brian (Narita family)

but then you have horses like Vodka and more that aren't from Japan so idrk. but they don't seem very Iterator-y.

To break down Iterator names:

Looks To The Moon - a phrase; "she looks to the moon" (Verb Preposition Article Noun)

Five Pebbles - Adjective Noun

No Significant Harassment - more of a phrase?? but also suits Adjective Adjective Noun

Seven Red Suns - Adjective Adjective Noun

Chasing / Grey Wind - Adjective Noun

Unparalleled Innocence - Adjective Noun

Sliver Of Straw - more phrase-like (Noun Preposition Noun)

so they seem to lean towards being a phrase or some form of adjective noun. but i think if you want to play with them you could take the structure of stuff like LTTM and SOS. I know I sure did (Legacy Of Famine and Thorns Without Roses). Have fun.

here's some i came up with:

“A Mountainous Tree, Fallen”

“Two Falling Blades Over an Ocean”

“A Feather or an Axe”

“Into the Sky”

“A Murder Forgiven”

“A Glade, an Effigy”

“The Clouds Have Burnt to Ashes”

“Skeleton of a Thunderstorm”

“Practices No Stoicism”